"Chapter 1 - The Op Amp's Place in the World" - HTL Wien 10

Sine Wave Oscillator Circuits
15-10
the output and Z 1, V TEST is applied to Z 1, and V OUT is calculated. The positive feedback
voltage, V +, is calculated first in Equations 15–6 through 15–8. Equation 15–6 shows the
simple voltage divider at the noninverting input. Each term is then multiplied by (R 2C 2s
+ 1) and divided by R 2 to get Equation 15–7.
V VTEST Z4 V
Z3 Z4 TEST
R2 R2C2s1 R 1 1
C1s V
V TEST
1
1 R1C2S R1 R2 1
R2C1s C2 C1 R 2
R 2 C 2 s1
(15–6)
(15–7)
Substitute s = jω 0, where ω 0 is the oscillation frequency, ω 1 = 1/R 1C 2, and ω 2 = 1/R 2C 1
to get Equation 15–8.
V
V TEST
1 R 1
R 2
1
C2 j C1 0
1 2 0 (15–8)
Some interesting relationships now become apparent. The capacitor in the zero, represented
by ω 1, and the capacitor in the pole, represented by ω 2, must each contribute 90
of phase shift toward the 180 required for oscillation at ω 0. This requires that C 1 = C 2 and
R 1 = R 2. Setting ω 1 and ω 2 equal to ω 0 cancels the frequency terms, ideally removing any
change in amplitude with frequency since the pole and zero negate one another. An overall
feedback factor of β = 1/3 is the result (Equation 15–9).
V
V TEST
1
1 R C
R C j 0 0
1
3 j 0
0 0
0
1
3
(15–9)
The gain of the negative feedback portion, A, of the circuit must then be set such that ⎟Aβ⎟
= 1, requiring A = 3. R F must be set to twice the value of R G to satisfy the condition. The
op amp in Figure 15–7 is single supply, so a dc reference voltage, V REF, must be applied
to bias the output for full-scale swing and minimal distortion. Applying V REF to the positive
input through R 2 restricts dc current flow to the negative feedback leg of the circuit. V REF
was set at 0.833V to bias the output at the midrail of the single supply, rail-to-rail input and
output amplifier, or 2.5 V. See Cahpter 4 for details on dc biasing single-supply op amps.
V REF is shorted to ground for split supply applications.
The final circuit is shown in Figure 15–8, with component values selected to provide an
oscillation frequency of ω 0 = 2πf 0, where f 0 = 1/(2πRC) = 15.9 kHz. The circuit oscillated
at 1.57 kHz due to slightly varying component values with 2% distortion. This high value
is due to the extensive clipping of the output signal at both supply rails, producing several